No idea what the tree is but would not remove. Middle photo shows Hypericum (St John's Wort). Bottom photo has a white daisy on the left, a pink rose and a wee white plant I can't i.d. I wouldn't get rid of any of them - the rose could do with pruning in autumn when flowering has finished.

A good rule of thumb in a new garden is not to do very much for the first year to see what appears.

Lucy, welcome to Goy, gardening and the UK. As you have already seen we can be quite helpful in identifying you plants for you but, being a virtual site, we are more limited in the general advice that we can give. I would suggest that you take a walk round your neighberhood and see if you can find a garden that is well looked after and has lots of perennial plants. Chat to the gardener and ask him/her to come and have a look at yours and advise. As Moongrower advises above, don't be in too much of a hurry to do anything drastic, you never know what bulbs lie beneath the surface.

the white flower on the left looks like one of the Leucanthemums and the smaller one on th right looks like Feverfew. Personally id remove the feverfew and keep the other daisy type.I hope you will ahve lots of bulbs , now hidden from view. Have you asked the neighours about the garden in spring.
the tree looks like one of the cherries ut dont know which one.